I don't have a lot to offer, that hasn't been said. Good advice above!
I would absolutely have a quick chat about the need for continuity between home and daycare. Kids can adapt to a certain amount of difference in routine, but that is pretty vast. And it's about half and half, between home and your place during the daytime.
I know it's tough, as mom is a friend, but I would try to be straight with her. If they want her in any kind of care environment, they HAVE to change something. Otherwise, they will have the same issue no matter where she goes.
PS: of course she didn't cry at grandma's house! Much like at home, I am sure, she was able to do what she wanted, when she wanted in terms of sleep and play. And if mom was using that as comparison; I think that is terribly unfair. Kids are totally different with parents/grandparents/aunts/etc than with "outsiders".